I don't understand the difference between tensor and tensor field.
I'm learning from Barret O'neill's Semi-Riemann Geometry and here are the definitions:
If $A:(V^*)^r \times V^s\to K$ transformation is $K$-multilinear then
$A$ is a tensor on $V$.$M$ is a manifold, $\mathfrak{X}(M)$ is the vector fields' set that is an
$F(M)$-module.If $A$ is
a tensor on $\mathfrak{X}(M)$ then we say $A$ is a tensor field on
$M$.
I did not understand the last sentence. What is the difference between a tensor and tensor field?
Best Answer
The difference is all in your head. Literally.
The difference in calling the same object $A$ a "tensor over $\mathfrak{X}(M)$" as opposed to "a tensor field over $M$" is that the former emphasizes the fact that we have an algebraic object: a tensor over some module, while the latter emphasizes the fact that underlying the module there is some manifold and geometry is going on there.
Calling something a tensor field instead of a tensor forces you to remember that $\mathfrak{X}(M)$ is not just some arbitrary module, but that its elements can be identified with smooth sections of the tangent bundle of some manifold. These additional structures are occasionally useful.